Airlock in cold tap! How do i fix it?

The cold water tap in the kitchen is usually the only one that comes directly off the mains. All others generally feed from a cold water tank in the attic. If you ran water from any tap other than the kitchen one when the mains was off, then the tank would empty. So eventually you would have no cold or hot water in your bathroom. As the pipes would also empty, that's when you can get an airlock (or a blockage as sludge from the tank bottom is drawn into the pipes.

First thing to check is see if the ball valve in the attic tank is dropped and letting the tank fill. If it isn't, waggle it up and down a few times to free it. If that's OK then try connecting the cold tap in the kitchen to the hot tap with a piece of hose or something, fully open the hot tap and then open the cold tap about one turn. Then in the attic put your thumb over then end of the open vent pipe over the tank -- that's normally a 22 mm copper pipe with an open end. Doing that should push any air or debris back into the tank and clear the system.